scholarly journals Analysis of a mass-spring-relay system with periodic forcing

2021 ◽  
Author(s):  
János Lelkes ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Fixed Points ◽  
Poincaré Map ◽  
Pitchfork Bifurcation ◽  
Global Dynamics ◽  
Poincare Map ◽  
Periodic Forcing ◽  
Relay System ◽  
Saddle Type ◽  
The Stability ◽  
Mass Spring

AbstractThe dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. An explicit Poincaré map is constructed with an implicit constraint on the switching time. The stability of the fixed points of the Poincaré map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle-type fixed point. The global dynamics of the system exhibits discontinuity induced bifurcations of the fixed points.

Analysis of a Mass-Spring-Relay System With Periodic Forcing

2021 ◽  
Author(s):  
János Lelkes ◽  
Tamás KALMÁR-NAGY
Keyword(s):  
Poincaré Map ◽  
Pitchfork Bifurcation ◽  
Global Dynamics ◽  
Poincare Map ◽  
Periodic Excitation ◽  
Periodic Forcing ◽  
Relay System ◽  
Saddle Type ◽  
The Stability ◽  
Mass Spring

Abstract The dynamics of a hysteretic relay oscillator with harmonic forcing is investigated. Periodic excitation of the system results in periodic, quasi-periodic, chaotic and unbounded behavior. A Poincare map is constructed to simplify the mathematical analysis. The stability of the xed points of the Poincare map corresponding to period-one solutions is investigated. By varying the forcing parameters, we observed a saddle-center and a pitchfork bifurcation of two centers and a saddle type xed point. The global dynamics of the system is investigated, showing discontinuity induced bifurcations of the xed points.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Computing the Number of Fixed Joints on Poincare Map in Nonlinear Mathieu Equation

1993 ◽  
Author(s):  
Hisato Fujisaka ◽  
Chikara Sato
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Numerical Method ◽  
Topological Degree ◽  
Poincaré Map ◽  
Mathieu Equation ◽  
Time Varying ◽  
Poincare Map ◽  
Newton's Iteration ◽  
Combined Use

Abstract A numerical method is presented to compute the number of fixed points of Poincare maps in ordinary differential equations including time varying equations. The method’s fundamental is to construct a map whose topological degree equals to the number of fixed points of a Poincare map on a given domain of Poincare section. Consequently, the computation procedure is simply computing the topological degree of the map. The combined use of this method and Newton’s iteration gives the locations of all the fixed points in the domain.

scholarly journals Poincaré Map Approach to Global Dynamics of the Integrated Pest Management Prey-Predator Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zhenzhen Shi ◽  
Qingjian Li ◽  
Weiming Li ◽  
Huidong Cheng
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Poincaré Map ◽  
Global Dynamics ◽  
Poincare Map ◽  
Existence And Stability ◽  
Sufficient And Necessary Conditions ◽  
Predator Model ◽  
Order One Periodic Solution

An integrated pest management prey-predator model with ratio-dependent and impulsive feedback control is investigated in this paper. Firstly, we determine the Poincaré map which is defined on the phase set and discuss its main properties including monotonicity, continuity, and discontinuity. Secondly, the existence and stability of the boundary order-one periodic solution are proved by the method of Poincaré map. According to the Poincaré map and related differential equation theory, the conditions of the existence and global stability of the order-one periodic solution are obtained when ΦyA<yA, and we prove the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution when ΦyA>yA. Furthermore, we prove the existence of the order-kk≥2 periodic solution under certain conditions. Finally, we verify the main results by numerical simulation.

scholarly journals Horseshoe Chaos in a Simple Memristive Circuit

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lei Wang ◽  
XiaoSong Yang ◽  
WenJie Hu ◽  
Quan Yuan
Keyword(s):  
Stability Analysis ◽  
Poincaré Map ◽  
Circuit Model ◽  
Computer Assisted ◽  
Poincaré Section ◽  
Poincare Map ◽  
Poincare Section ◽  
Topological Horseshoe ◽  
Memristive Circuit ◽  
The Stability

A simple memristive circuit model is revisited and the stability analysis is to be given. Furthermore, we resort to Poincaré section and Poincaré map technique and present rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoe theory.

Stability and Bifurcation Analysis of an Asymmetrically Electrostatically Actuated Microbeam

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Hadi Madinei ◽  
Ghader Rezazadeh ◽  
Saber Azizi
Keyword(s):  
Applied Voltage ◽  
Equilibrium Points ◽  
Pitchfork Bifurcation ◽  
Spring Model ◽  
Electrostatically Actuated ◽  
Proposed Model ◽  
Mass Spring Model ◽  
The Stability ◽  
Mass Spring ◽  
Two Degree Of Freedom

This paper deals with the study of bifurcational behavior of a capacitive microbeam actuated by asymmetrically located electrodes in the upper and lower sides of the microbeam. A distributed and a modified two degree of freedom (DOF) mass–spring model have been implemented for the analysis of the microbeam behavior. Fixed or equilibrium points of the microbeam have been obtained and have been shown that with variation of the applied voltage as a control parameter the number of equilibrium points is changed. The stability of the fixed points has been investigated by Jacobian matrix of system in the two DOF mass–spring model. Pull-in or critical values of the applied voltage leading to qualitative changes in the microbeam behavior have been obtained and has been shown that the proposed model has a tendency to a static instability by undergoing a pitchfork bifurcation whereas classic capacitive microbeams cease to have stability by undergoing to a saddle node bifurcation.

HORSESHOES IN THE TWIST-AND-FLIP MAP

1991 ◽  
Vol 01 (01) ◽  
pp. 235-252 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA
Keyword(s):  
Poincaré Map ◽  
Closed Form Expression ◽  
Analytical Relationship ◽  
Sufficient Condition ◽  
Poincare Map ◽  
Form Expression ◽  
Periodic Forcing ◽  
Nonlinear Odes ◽  
Square Wave ◽  
Analytical Test

We derive an analytical relationship between the parameters of a square-wave forced, nonlinear, two-dimensional ordinary differential equation which determines conditions under which the Poincaré map has a horseshoe. This provides an analytical test for chaos for this equation. In doing this we show that the Poincaré map has a closed-form expression as a transformation of R2 of the form FTFT, where F is a flip, i.e., a 180-degree rotation about the origin and T is a twist centered at (a, 0) for a > 0. We show that this derivation is quite general. We aiso show how to relate our results to ODEs with continuous periodic forcing (e.g., the sinusoidal-forced Duffing equation). Finally, we provide a conjecture as to a sufficient condition for chaos in square-wave forced, nonlinear ODEs.

scholarly journals STUDYING THE BASIN OF CONVERGENCE OF METHODS FOR COMPUTING PERIODIC ORBITS

2011 ◽  
Vol 21 (08) ◽  
pp. 2079-2106 ◽  
Author(s):  
MICHAEL G. EPITROPAKIS ◽  
MICHAEL N. VRAHATIS
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Newton's Method ◽  
Poincaré Map ◽  
Specific Problem ◽  
Poincare Map ◽  
Robust Methods ◽  
Surface Of Section ◽  
Cubic Equation ◽  
Nonlinear Mappings

Starting from the well-known Newton's fractal which is formed by the basin of convergence of Newton's method applied to a cubic equation in one variable in the field ℂ, we were able to find methods for which the corresponding basins of convergence do not exhibit a fractal-like structure. Using this approach we are able to distinguish reliable and robust methods for tackling a specific problem. Also, our approach is illustrated here for methods for computing periodic orbits of nonlinear mappings as well as for fixed points of the Poincaré map on a surface of section.

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